Content-Type: multipart/mixed; boundary="-------------0306150918745" This is a multi-part message in MIME format. ---------------0306150918745 Content-Type: text/plain; name="03-283.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-283.comments" PACS 75.50.Lk ---------------0306150918745 Content-Type: text/plain; name="03-283.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-283.keywords" Spin glass, SK model, replica symmetry breaking, ultrametricity, variational bounds, GREM, Sherrington Kirkpatrick model, Parisi solution, Guerra bounds, rigorous results, critical review ---------------0306150918745 Content-Type: application/x-tex; name="SK_uploaded.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="SK_uploaded.tex" %% DATE: June 14, 2003 %\documentstyle[preprint,aps]{revtex} % FOR TWO COLUMN ACTIVATE THE LINE BELOW %%% %%% revtex4. %%% \documentclass[prl,draft,showpacs,twocolumn]{revtex4} \usepackage{amsmath,amsfonts,amsthm} % \input epsf %preferred figures program % % Theorems ---------------------------------------------------- % \newtheorem{thm}{Theorem}[section] \newtheorem{lem}{Lemma}[section] \def\N{\mathbb N} \def\E{\mathbb E} %%% expectation \def\qed{$\Box$} \def\sig{ \sigma} \newcommand{\eq}[1]{eq.~(\ref{#1})} %% produces: eq.(#) \newcommand{\Ev}[1]{\E \left( #1 \right)} %% produces \E( # ) \newcommand{\Er}[1]{\E^{(2)} \left( #1 \right)} %% produces \E^2( # ) \newcommand{\Ert}[1]{\E^{(2)}_t \left( #1 \right)} %% produces \E^2_t( # ) \newcommand{\Deq}{\atop{\mathcal{D}}{=}} \def\eps{\varepsilon} \def\K{{\kappa}} \begin{document} % FOR TWO COLUMN ACTIVATE THE LINE BELOW %\twocolumn[\hsize\textwidth\columnwidth\hsize\csname %@twocolumnfalse\endcsname %%% %%% \draft is now invoked as an option %%% %\draft \title{An Extended Variational Principle for the SK Spin-Glass Model} % \draft command makes pacs numbers print % repeat the \author\address pair as needed \author{Michael Aizenman, Robert Sims, and Shannon L. Starr} %%% %%% Changed \address to \affiliation %%% \affiliation{ Departments of Physics and Mathematics, Jadwin Hall, Princeton University, Princeton, NJ 08544} \date{\today} \begin{abstract} The recent proof by F. Guerra that the Parisi ansatz provides a lower bound on the free energy of the SK spin-glass model could have been taken as offering some support to the validity of the purported solution. In this work we present a broader variational principle, in which the lower bound, as well as the actual value, are expressed through an optimization procedure for which ultrametic/hierarchal structures form only a subset of the variational class. The validity of Parisi's ansatz for the SK model is still in question. The new variational principle may be of help in critical review of the issue. \end{abstract} \pacs{75.50.Lk} %%%%%%% \maketitle %%%%%%% \noindent{\em Introduction \qquad } The statistical mechanics of spin-glass models is characterized by the existence of a diverse collection of competing states, very slow relaxation of the quenched dynamics, and a rather involved picture of the equilibrium state. A great deal of insight on the subject has been produced through the study of the Sherrington Kirkpatrick (SK) model~\cite{SK}. After some initial attempts, a solution was proposed by G. Parisi which has the requisite stability and many other attractive features~\cite{P}. Its development has yielded a plethora of applications of the method, in which a key structural assumption is a particular form of the replica symmetry-breaking (i.e., the assumption of ``ultrametricity'', or the hierarchal structure, of the overlaps among the observed spin configurations)~\cite{MPV}. Yet to this day it was not established that this very appealing proposal does indeed provide the equilibrium structure of the SK model. A recent breakthrough is the proof by F.~Guerra~\cite{Gue} that the free energy provided by Parisi's purported solution is a rigorous lower bound for the SK free energy. More completely, the result of Guerra is that for any value of the order parameter, which within the assumed ansatz is a {\em function}, the Parisi functional provides a rigorous lower bound. Thus, this relation is also valid for the maximizer which yields the Parisi solution. In this work we present a variational principle for the free energy of the SK model which makes no use of a Parisi-type order parameter, and which yields the result of Guerra as a particular implication. More explicitly, the new principle allows more varied bounds on the free energy, for which there is no need to assume a hierarchal organization of the Gibbs state (e.g., as expressed in the assumed ultrametricity of the overlaps~\cite{MPV}). Guerra's results follow when the variational principle is tested against the Derrida-Ruelle hierarchal probability cascade models (GREM)~\cite{Ru}. This leads us to a question which is not new: is the ultrametricity an inherent structue of the SK mean-field model, or is it only a simplifiying assumption. The new variational principle may provide a tool for challenging tests of this issue. \noindent{\em The model \qquad} The SK model concerns Ising-type spins, $\sig = (\sigma_1,\dots,\sigma_N)$, with an a-priori equi-distribution over the values $\{ \pm 1\}$, and the random Hamiltonian \begin{equation} H_{N}(\sig)\ = \ \frac{-1}{\sqrt{N} } \sum_{1\le i < j \le N} J_{ij} \, \sigma_{i}\sigma_{j} \ - \ h \sum_{i=1}^N \sigma_{i} \end{equation} where $\{J_{ij}\}$ are independent normal Gaussian variables. Our analysis applies to a more general class of Hamiltonians which includes all the even ``$p$-spin'' models~\cite{Der,GT2}. Namely: \begin{equation} H_N(\sig)\ = \ - K_N(\sig) \ - \ h \sum_{i=1}^N \sigma_{i} \end{equation} with \begin{equation} \label{eq:pspin} K_N(\sig)\ = \ \sqrt{ \frac{N}{2} } \, \sum_{r=1}^\infty \frac{a_r}{ N^{r/2}} \sum_{i_1,\dots,i_r=1}^N J_{i_1\dots i_r} \sigma_{i_1}\cdots\sigma_{i_r} \end{equation} where all the $\{J_{i_1,\dots,i_r}\}$ are independent normal Gaussian variables (for convenience, the tensor is not assumed here to be symmetric), and $\sum_{r=1}^{\infty} |a_r|^2 =1$. As in \cite{GT2}, our argument requires that the function $ f(q) \ = \ \sum_{r=1}^{\infty} |a_r|^2 \, q^r \, $ be convex on $[-1,1]$. One may note that $K_N(\sig)$ form a family of centered Gaussian variables with the covariance \begin{equation} \Ev{ K_N(\sig)\, K_N(\sig')\ \vert\ \sig\, ,\, \sig'} \ = \ \frac{N}{2} \, f( q_{\sig,\sig'}) \, , \label{eq:fq} \end{equation} which depends on the spin-spin overlap: $q_{\sig, \sig'} = \frac{1}{N} \sum_j \sigma_j \sigma'_j$. The standard SK model corresponds to $f(q)=q^2$. The partition function, $Z$, the quenched free energy, $F$, and what we shall call here the pressure, $P$, are defined as \begin{gather} Z_{N}(\beta,h) = \sum_{\sigma_1,\ldots, \sigma_N = \pm 1} e^{-\beta\ H_{N}(\sig)} \label{eq:Z} \\ P_{N}(\beta,h) \ = \ \frac{1}{N} \Ev{ \log Z_{N}(\beta,h) } \ = \ - \beta\ F_{N}(\beta,h) \end{gather} where $\Ev{-}$ is an average over the random couplings $\{J_{ij}\}$. The thermodynamic limit for the free energy, i.e., the existence of $\lim_{N\to \infty}P_{N}(\beta,h) = P(\beta,h)$, was recently established by Guerra-Toninelli~\cite{GT} through a much-awaited argument. \medskip \noindent{\em The Variational Principle \quad } Our variational expression for $P(\beta,h)$ employs a setup which may at first appear strange, but is natural from the cavity perspective, when one considers the change in the total free energy caused by the addition of $M$ spins to a much larger system of size $N$. The expression for $Z_{N+M}/Z_{N}$ simplifies in the limit $N\to \infty$, at fixed $M$. In the following idealized definition one may regard the symbol $\alpha$ as representing the configuration of the bulk. The discretness seen in the definition ($\sum \xi_{\alpha}$) is just for the convenience of the formulation of the variational bounds, and not an assumption on the Gibbs state, though such an assumption may well be true. (A more general formulation is possible, but not much is lost by restricting attention to the ``ROSt'' defined below.) % \begin{defn} \noindent {\bf Definition (Random Overlap Structures):} { A {\em random overlap structure} (ROSt) consists of a probability space $\{\Omega, \mu(d\omega) \} $ where for each $\omega$ there is associated a system of weights $\{ \xi_{\alpha}(\omega)\}$ and an ``overlap kernel'' $\{ \tilde{q}_{\alpha, \alpha'}(\omega) \} $ such that, for each $\omega \in \Omega$, \begin{enumerate} \item[i.] $\sum_{\alpha} \xi_{\alpha}(\omega) \ \le \ \infty $, \item[ii.] the quadratic form corresponding to $\{ \tilde{q}_{\alpha, \alpha'}\}$ is positive definite, \item[iii.] $\tilde{q}_{\alpha, \alpha}=1$, for each $\alpha$, and hence (by the Schwarz inequality) also: $|\tilde{q}_{\alpha, \alpha'}|\le 1$ for all pairs $\{ \alpha, \alpha'\}$. \end{enumerate} } % \end{defn} An important class of ROSt's is provided by the Derrida-Ruelle probability cascade model which is formulated in ref.~\cite{Ru} (called there GREM). Without additional assumptions, one may associate to the points in any ROSt two independent families of centered Gaussian variables $\{\eta_{j,\alpha} \}_{j=1,2,\ldots}$ and $\{\K_{\alpha}\}$ with covariances (conditioned on the random configuration of weights and overlaps) \begin{gather} \label{eq:etacov} \Ev{\, \eta_{j,\alpha} \eta_{j',\alpha'} \, | \, \tilde{q}_{\alpha, \alpha'} \, } \ = \ \frac{1}{2} \delta_{j,j'} \ f'(\tilde{q}_{\alpha, \alpha'}) \, ,\\ \label{eq:kapcov} \Ev{\, \K_{\alpha} \K_{\alpha'} \, | \, \tilde{q}_{\alpha, \alpha'} \, } \ = \ \tilde{q}_{\alpha, \alpha'} f'(\tilde{q}_{\alpha, \alpha'}) - f(\tilde{q}_{\alpha, \alpha'}) \, . \end{gather} The existence of such processes requires positive-definiteness of the joint covariance, but that is evident from the following explicit construction in the case that the $\alpha$'s are $N$-vectors, with $q_{\alpha,\alpha'}=\frac{1}{N}\sum_j \alpha_j,\alpha_{j}'$: \begin{equation} \eta_{j,\alpha} \ = \ \sqrt{\frac{N}{2}} \sum_r \frac{\sqrt{r} \, a_r}{N^{r/2}} \sum \, \widetilde{J}_{j,i_1,\ldots,i_{r-1}} \alpha_{i_1}\cdots\alpha_{i_{r-1}} \end{equation} where the second sum is over $i_1,\ldots,i_{r-1}$ which range from $1$ to $N$, and \begin{equation} \K_{\alpha} \ = \ \sum_r \frac{\sqrt{r-1}\, a_r}{N^{r/2}} \sum_{i_1,\ldots,i_r=1}^{N} \, \widehat{J}_{i_1,\ldots,i_r} \alpha_{i_1}\cdots\alpha_{i_r} \end{equation} We shall now denote by $\E(\cdot)$ the combined average, which corresponds to integrating over three sources of randomness: the SK random couplings $\{ J_{ij} \}$, the random overlap structure described by the measure $\mu(d\omega)$, and the Gaussian variables $\{\K_{\alpha}\}$ and $\{ \eta_{j,\alpha} \}$. Guided by the cavity picture, we associate with each ROSt the following quantity: \begin{multline} \label{ew:G} G_M(\beta,h; \mu) \ = \\ = \frac{1}{M} \ \Ev{ \, \log \left( \frac{\sum_{\alpha, \sig} \xi_{\alpha} \, e^{\beta \sum_{j=1}^{M} (\eta_{j,\alpha} + h ) \sigma_{j} } } {\sum_{\alpha} \xi_{\alpha} \, \, e^{\beta \sqrt{M/2} \, \K_{\alpha} } }\right) } \end{multline} where $\sig = (\sigma_1,\ldots, \sigma_M) $ Our main result is: % \begin{thm} \noindent{\bf Theorem 1} {\em {\em i.} For any finite $M$, \begin{equation} P_{M}(\beta,h) \ \le \ \inf_{(\Omega, \mu)} \ G_M(\beta,h; \mu) \ \le \ P_{U}(\beta,h) \, , \label{eq:AFN} \end{equation} where the infimum is over random overlap structures (ROSt's) and $P_{U}(\beta,h)$ denotes the free energy $\times(-\beta) $ obtained through the Parisi ``ultrametric'' (or ``hierarchal'') ansatz. \\ {\em ii.} The infinite volume limit of the free energy satisfies: \begin{equation} \label{eq:BFN} P(\beta,h) \ = \ \lim_{M\to \infty} \inf_{(\Omega, \mu)} \ G_M(\beta,h; \mu) \, \, . \end{equation} } % \end{thm} % \begin{proof} \noindent{\em Proof \quad } These results can be seen as consisting of two separate parts: lower and an upper bounds, which are derived by different arguments. {\em i.} The upper bound: the left inequality in \eq{eq:AFN}, employs an interpolation argument which is akin to that used in the analysis of Guerra~\cite{Gue}, but which here is formulated in broader terms without invoking the ultrametric ansatz. The second inequality in (\ref{eq:AFN}) holds since the Parisi calculation represents the restriction of the variation to the subset of hierarchal ROSt's. To derive the first inequality let us introduce a family of Hamiltonians for a mixed sytem of $M$ spins $\sig = (\sigma_1,\dots,\sigma_M)$ and the ROSt variables $\alpha$, with a parameter $0\le t\le 1$: \begin{multline} - H_{M}(\sig, \alpha; t) = \frac{\sqrt{(1-t)M}}{\sqrt{2}}(K_{M}(\sig) + \K_\alpha)\ + \\ +\ \sqrt{t} \sum_{j=1}^{M} \eta_{j,\alpha} \sigma_{j} \ +\ h \sum_{j=1}^M \sigma_j\, , \end{multline} and let \begin{equation} P_M(\beta,h;t) \ = \ \frac{1}{M} \ \Ev{ \, \log \left( \frac{\sum_{\alpha, \sig} \xi_{\alpha} \, e^{ - \beta H_{M}(\sig, \alpha; t) } } {\sum_{\alpha} \xi_{\alpha} \, \, e^{\beta \sqrt{M/2} \, \K_{\alpha} } } } \right)\, . \end{equation} Then \begin{align} P_M(\beta,h;0) &= P_M(\beta,h)\, , \label{eq:int0} \\ P_M(\beta,h;1) &= G_M(\beta,h; \mu)\, , \label{eq:int1} \end{align} and we shall show that $ \frac{d}{dt}P_M(\beta,h;t) \ \ge \ 0$. We use the following notation for replica averages over pairs of spin and ROSt variables. For any $X=X(\sig,\alpha)$ and $Y=Y(\sig,\alpha;\sig',\alpha')$: \begin{eqnarray} \E^{(1)}_{t}(X) & = &\Ev{\ \sum_{\alpha,\sig} \ w(\sig,\alpha;t)\ X \ } \\ \Ert{Y} \ & = & \ \Ev{\ \sum_{\alpha,\sig} \sum_{\alpha',\sig'}\ w(\sig,\alpha;t)\ w(\sig',\alpha';t) \ Y \ } \nonumber \end{eqnarray} with the ``Gibbs weights'' \begin{equation} w(\sig,\alpha;t) = \xi_{\alpha} e^{-\beta H_{M}(\sig,\alpha;t)} \Big/ \sum_{\alpha,\sig} \xi_{\alpha} e^{-\beta H_{M}(\sig,\alpha;t)} \, . \end{equation} We now have \begin{equation} \frac{d}{dt}P_M(\beta,h;t) \ = - \frac{\beta}{M} \E_t^{(1)}\left( \frac{d}{dt} H_M(\sig,\alpha;t) \right) \, . \end{equation} The term $\frac{d}{dt} H_M(\sig,\alpha;t)$ includes Gaussian variables, and one may apply to it the generalized Wick's formula (Gaussian integration-by-parts) for correlated Gaussian variables, $x_1,\dots,x_n$: \begin{multline} \langle x_1\, \psi(x_1,\dots,x_n) \rangle \ = \\ = \ \sum_{j=1}^n \langle x_1 x_j \rangle\, \langle \frac{\partial \psi(x_1,\dots,x_n)}{\partial x_j} \rangle \, . \end{multline} The result is: \begin{equation} - \frac{\beta}{M} \E_t^{(1)}\left( \frac{d}{dt} H_M(\sig,\alpha;t) \right)\ = \ \frac{\beta^2}{4}\, \E_t^{(2)}\left( \varphi \right) \, \end{equation} where \begin{multline} \varphi(\sig,\alpha;\sig',\alpha')\ =\\ = \ \frac{d}{dt} \Ev{(H_M(\sig,\alpha;t) - H_M(\sig',\alpha';t))^2 \Big\vert \tilde{q}_{\alpha,\alpha'}, q_{\sig,\sig' }} \\ = \ [f(q_{\sig,\sig'}) - f(\tilde{q}_{\alpha,\alpha'})] - (q_{\sig,\sig'} - \tilde{q}_{\alpha,\alpha'}) f'(\tilde{q}_{\alpha,\alpha'}) \, . \end{multline} Therefore, \begin{eqnarray} \frac{d}{dt}P_M(\beta,h;t) \ = \ \frac{\beta^2}{4} \ \times \ \qquad \qquad \qquad \qquad \qquad \qquad \\ \Ert{[f(q_{\sig,\sig'}) - f(\tilde{q}_{\alpha,\alpha'})] - (q_{\sig,\sig'} - \tilde{q}_{\alpha,\alpha'}) f'(\tilde{q}_{\alpha,\alpha'}) } \ge 0 \nonumber \, . \end{eqnarray} The last inequality, which is crucial for us, follows from the assumed convexity of $f$. For the SK model, the above expression simplifies to $\Ert{ (q_{\sig,\sig'}-\tilde{q}_{\alpha, \alpha'} )^{2} } $. Putting the positivity of the derivative together with (\ref{eq:int0}) and (\ref{eq:int1}) clearly implies the first bound in (\ref{eq:AFN}). As was noted earlier, a particular class of random overlap structures is provided by the Derrida-Ruelle probability cascade models (GREM) of~\cite{Ru}, which are parametrized by a monotone function $x: [0,1] \to [0,1]$. These models have two nice features: {\em i.} the distribution of $\{\xi_{\alpha}\}$ is invariant, except for a deterministic scaling factor, under the multiplication by random factors as in (\ref{ew:G}) (consequently the value of $G_M(\ldots,\mu_{x(\cdot)})$ for such ROSt does not depend on $M$), {\em ii.} quantities like $G_M(\ldots,\mu_{x(\cdot)})$ can be expressed as the boundary values of the solution of a certain differential equation, which depends on $x(\cdot)$. Evaluated for such models $G_M(\ldots,\mu_{x(\cdot)})$ reproduces the Parisi functional for each value of the order parameter $x(\cdot)$. The Parisi solution is obtained by optimizing (taking the {\em inf}) over the order parameter $x(\cdot)$. This relation gives rise to the second inequality in (\ref{eq:AFN}). {\em ii.} To prove (\ref{eq:BFN}) we need to supplement the first inequality in (\ref{eq:AFN}) by an opposite bound. Our analysis is streamlined by continuity arguments, which are enabled by the following basic estimate (proven by two elementary applications of the Jensen inequality). \noindent{\bf Lemma 2} {\em Let $Z(H)$ denote the partition function for a system with the Hamiltonian $H(\sigma)$, and let $U(\sigma)$ be, for each $\sigma$, a centered Gaussian variable which is independent of $H$. Then \begin{equation} \label{Pb} 0 \ \le \ \E\left(\log \frac{Z(H+U)}{Z(H)} \right) \ \le \ \frac{1}{2}\, \E(\, U^2 \, ) \, . \end{equation} } Using the above, it suffices to derive our result for interactions with the sum over $r$, in (\ref{eq:pspin}), truncated at some finite value. A convenient tool is provided by the superadditivity of $Q_N \equiv N\, P_N$, which was established in the work of Guerra-Toninelli~\cite{GT} and its extensions~\cite{GT2,CDGG}. The statement is that for the systems discussed here (and in fact a broader class) for each $M, N \in \N$ \begin{equation} \label{eq:superadd} Q_{M+N}(\beta,h) \geq Q_M(\beta,h) + Q_N(\beta,h)\, . \end{equation} The superadditivity was used in \cite{GT} to establish the existence of the limit $\lim_{N\to \infty} P_N$. However, it has a further implication based on the following useful fact. \noindent{\bf Lemma 3 } {\em For any superadditive sequence $\{Q_N\}$ satisfying (\ref{eq:superadd}) the following limits exist and satisfy \begin{equation} \lim_{N \to \infty} \, Q_N / N = \lim_{M \to \infty} \liminf_{N \to \infty}\, \, [Q_{M+N} - Q_N]/M \, . \end{equation} } For our purposes, this yields: \begin{equation} \lim_{N \to \infty} P_N \ = \ \lim_{M \to \infty} \liminf_{N \to \infty}\, \frac{1}{M} \, \E\left(\log \frac{Z_{N+M}}{Z_{N}} \right) \, . \end{equation} We now claim, based on an argument employing the cavity picture, that for any $M$ \begin{multline} \label{eq:Opp} \liminf_{N \to \infty} \frac{1}{M} \E\left(\log \frac{Z_{N+M}}{Z_{N}} \right) \ \ge \ \inf_{(\Omega,\mu)} G_M(\beta,h;\mu) \, , \end{multline} which would clearly imply (\ref{eq:BFN}). The reason for this inequality is that when a block of $M$ spins is added to a much larger ``reservoir'' of $N$ spins, the change in the free energy is exactly in the form of (\ref{ew:G}) -- except for corrections whose total contribution to $G_M$ is of order $O(\frac{M}{N})$. % (due to the (The spin-spin couplings within the smaller block and the subleading terms from the change $N \mapsto (N+M)$ in (\ref{eq:pspin}).) Thus, the larger block of spins acts as a ROSt. To see that in detail, let us split the system of $M+N$ spins into $\sigma = ( \tilde{\sigma}, \alpha)$, with $\tilde{\sigma}= (\sigma_1,\dots,\sigma_M )$ and $\alpha = (\sigma_{M+1},\dots,\sigma_{M+N} )$. With this notation, the interaction decomposes into \begin{equation} K_{M+N}(\sig) \ = \ \widetilde{K}_{N}(\alpha) + \sum_{j=1}^M \widetilde{\eta}_{j,\alpha} \, \sigma_j +U(\tilde{\sigma}, \alpha) \end{equation} where: {\em i.} $\{\widetilde{K}_N(\alpha) \}$ consists of the terms of $K_{M+N}(\sig) $ which involve only spins in the larger block, {\em ii.} the second summand includes all the terms which involve exactly one spin in the smaller block, and {\em iii.} $U$ consists of the remaining terms of $K_{M+N}(\sig) $, including the spin-spin interactions within the smaller block. One should note that $\{\widetilde{K}_N(\alpha) \} \neq \{ K_N(\alpha) \}$ since, as a consequence of the addition of the smaller block, the terms in $\{\widetilde{K}_N(\alpha) \}$ are weighted by powers of $(N+M)$ rather than $N$, as presented in (\ref{eq:pspin}). By the law of addition of independent Gaussian variables, $ \{ K_N(\alpha) \}$ (which are of higher variance than $ \{ \widetilde{K}_N(\alpha) \}$ ) have the same distribution as the sum of independent variables \begin{equation} \left\{ \widetilde{K}_N( \alpha) + \sqrt{ \frac{M}{2}}\kappa_{\alpha} \right\} \, , \end{equation} where $\{ \kappa_{\alpha} \}$ are centered Gaussian variables independent of $\widetilde{K}_N( \alpha) $. Up to factors $[1+O( \frac{M}{N})]$, the covariances of $\{\widetilde{\eta}_{j, \alpha} \}$ and $\{ \kappa_{\alpha} \}$ satisfy (\ref{eq:etacov}) and (\ref{eq:kapcov}), respectively, and \begin{equation} \label{Ub} \frac{1}{M}\, \E ( U( \tilde{\sigma}, \alpha)^2 ) \le C \frac{M}{N} \, . \end{equation} Taking \begin{equation} \xi_{\alpha} := \mbox{ exp} \left[ \beta \left( \widetilde{K}_N(\alpha)+ h \sum_{i=1}^{N} \alpha_i \right) \right], \end{equation} we find that (\ref{eq:Opp}) follows by directly substituting the above into (\ref{ew:G}) (using (\ref{Pb}) and (\ref{Ub})). \qed \noindent{\em Discussion \/ } At first glance, the recent result of \cite{Gue} may be read as offering some support to the widely shared belief that the Parisi ansatz has indeed provided the solution of the SK model. However, we showed here that the Guerra bound is part of a broader variational principle in which no reference is made to the key assumption of \cite{P} that in the limit $N\to \infty$ the SK Gibbs state develops a hierarchal organization. The reasons for such an organization, which is equivalently expressed in terms of ``ultrametricity'' in the overlaps $q_{\sig,\sig'}$, are not a-priori obvious. (A step, approaching the issue from a dynamical perspective, was taken in ref~\cite{RA}, but this result has yet to be extended to the interactive cavity evolution.) Our result (\ref{eq:AFN}) raises the possibility that perhaps some other organizing principles may lead to even lower upper-bounds. This reinstates the question whether the ultrametricity assumption, which has enabled the calculation of \cite{P}, is correct in the context of the SK-type models. It should be emphasized, however, that the question is not whether the SK model exhibits replica symmetry breaking at low temperatures. That, as well as many other aspects of the accepted picture, are supported by both intuition and by rigorous results (\cite{ALR,FZ,PS,Tal,NS}). The question concerns the validity of a solution-facilitating ansatz about the hierarchal form of the replica symmetry breaking. The interest in this question is enhanced by fact that this assumption yields a computational tool with many other applications~\cite{MPV}. \noindent{\em Acknowledgements \/ } This work was supported, in part, by NSF grant PHY-9971149. R. Sims and S. Starr gratefully acknowledge the support of NSF Postdoctoral Fellowships (MSPRF). % \bibliography{SK} % to rerun bibtex % \bibliographystyle{apsrev} % \end{document} % comment out when done \begin{thebibliography}{15} \expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi \expandafter\ifx\csname bibnamefont\endcsname\relax \def\bibnamefont#1{#1}\fi \expandafter\ifx\csname bibfnamefont\endcsname\relax \def\bibfnamefont#1{#1}\fi \expandafter\ifx\csname citenamefont\endcsname\relax \def\citenamefont#1{#1}\fi \expandafter\ifx\csname url\endcsname\relax \def\url#1{\texttt{#1}}\fi \expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi \providecommand{\bibinfo}[2]{#2} \providecommand{\eprint}[2][]{\url{#2}} \bibitem[{\citenamefont{Sherrington and Kirkpatrick}(1975)}]{SK} \bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Sherrington}} \bibnamefont{and} \bibinfo{author}{\bibfnamefont{S.}~\bibnamefont{Kirkpatrick}}, \bibinfo{journal}{Phys. Rev. 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